arXiv research

In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…

In this paper, we explore the similarity between normal homogeneity and $δ$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $δ$-homogeneous Finsler space is $G$-$δ$-homo-geneous, for some suitably chosen connected quasi-compact $G$. So $δ$-homogeneous Fins…

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous sp…

The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.

In this paper we develop new methods of study of generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres. We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$, the fami…

New formula for Lichnerowicz Laplacian on homogeneous spaces.

We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut $δ$-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any $δ$-homogeneous Riemannian manifolds is itself $δ$-homogeneous. In turn, every simply connecte…

Finite extinction time for Hermitian curvature flow on homogeneous spaces.

We prove that a normal homogeneous space with the property that every Jacobi field along a geodesic vanishing at two points is the restriction of a Killing field along that geodesic is a globally symmetric space.

Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …

Study shows algebraic nature of manifold submetries on compact spaces.

Characterizes connections on normal distributions manifold.

Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.

We study the motion of charged particle under a natural choice of electromagnetic field in a general class of compact homogeneous spaces. As a special case we describe the motion in homogeneous Riemannian spaces $(G/H,g)$, where $g$ is any deformation of a normal metric along the fibers of a homogeneous fibration $K/H\…

Expanding on subgroup dimension, this paper covers new dimensions and compactness.

We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundl…

In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that…

Normal forms and moduli stacks for flat connections on complex manifolds.

Study on irreducibility of Laplacian eigenspaces in homogeneous spaces.

Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal ho…

Study Ricci flow on specific homogeneous spaces.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

Study harmonicity of normal almost contact structures on Riemannian manifolds.

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula wh…

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral repre…

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is not transitive (on t…

New extensions for homogeneous distributions on deformations to the normal cone.

We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.

Classifies semisimple symmetric contact spaces under Lie groups.

The paper studies geodesic orbit properties in Finsler spaces.

We study riemannian coverings $\varphi: \widetilde{M} \to Γ\backslash \widetilde{M}$ where $\widetilde{M}$ is a normal homogeneous space $G/K_1$ fibered over another normal homogeneous space $M = G/K$ and $K$ is locally isomorphic to a nontrivial product $K_1\times K_2$. The most familiar such fibrations $π: \widetilde…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

The authors give a short survey of previous results on $δ$-homogeneous Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these resul…

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary differential equations, respectively. We present here the qualitative study of these sys…

The paper explores new metrics on Lie groups and their geodesic properties.

Study on homogeneous geodesics in sub-Riemannian geometry.

New connections on symmetric spaces with invariant properties.

The study of submanifolds with constant principal curvatures in symmetric spaces.

Researchers discover all affinely homogeneous models for surfaces in 4D space.

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor ret…

The paper analyzes systoles of complex projective spaces under various metrics.

Study on rolling Stiefel manifolds with specific metrics.

Let $(M,F)$ be a connected Finsler space. An isometry of $(M,F)$ is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space $(M,F)$ is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close…

We generalize Sunada's method to produce new examples of closed, locally non-isometric manifolds which are isospectral. In particular, we produce pairs of isospectral, simply-connected, locally non-isometric normal homogeneous spaces. These pairs also allow us to see that in general group actions with discrete spectra …

The study verifies a conjecture about homogeneous quotients of manifolds with positive curvature.